Gaussian free fields at the integer quantum Hall plateau transition

Abstract

In this work we put forward an effective Gaussian free field description of critical wavefunctions at the transition between plateaus of the integer quantum Hall effect. To this end, we expound our earlier proposal that powers of critical wave intensities prepared via point contacts behave as pure scaling fields obeying an Abelian operator product expansion. Our arguments employ the framework of conformal field theory and, in particular, lead to a multifractality spectrum which is parabolic. We also derive a number of old and new identities that hold exactly at the lattice level and hinge on the correspondence between the Chalker-Coddington network model and a supersymmetric vertex model.Comment: 48 pages, 4 figure

On truncations of the Chalker-Coddington model

“…On the other hand, it is remarkable that the structure of our theory is qualitatively similar to the one in [9]. Like these authors, we find a normalizable ground state separated by a finite gap from a continuum, and, when restricting to pure winding or pure watermelon operators, two families of basic observables corresponding to su(2) spins and the continuous sl(2, R) series.…”

“…We note meanwhile that a subset of the exponents for the critical theory in the untruncated case was conjectured in [9] to be x CC = ∆ +∆ = Xp(p + 1) + Xq(1 − q); p = 0, 1, 2 . .…”

“…where X ≈ 1 4 . This formula corresponds to exponents proportional to the quadratic Casimir of gl(2|2), the number p corresponding to integer discrete spins j 1 = p of su(2), the number q to the principal continuous series of sl(2, R) (in [9] the fact that p is integer is a consequence of the Hilbert space of the spin chain being of the form (V ⊗ V * ) ⊗L . Other choices (such as an extra V or V * ) would allow p half-integer as well [40]).…”

“…More recently, a considerable effort has been devoted to better understand the nature of the lattice observables in relation with their continuum limit, making heavy use of symmetry considerations and the simple but very fruitful idea that wave intensities obey an operator product expansion of Abelian form. Abandoning speculation on the sigma model description of the full critical theory, the authors in [8,9] have strongly advocated a Gaussian free field description of critical wave functions at the transition, with, in particular, a parabolic spectrum exactly of the form (2)-even though the identification of the theory as a "simple" P SL(2|2) WZNW model is unlikely. Thorough numerical evidence seems to support this claim.…”

“…Conclusions are presented in section 9. In two appendices we discuss the interpretation of the continuum limit of the integrable modified truncated model as a sigma model, as well as the possible geometrical interpretation of scaling observables defined in the recent work [8,9].…”